Data Analysis and Related Applications 3

5An Evaluation of the Efficiency of a Shape Parameter Estimator for the Log-logistic Distribution

The log-logistic distribution has been successfully used in various fields. This model is often presented with shape and scale parameters. For additional flexibility, a location parameter can be added resulting in the three-parameter or shifted log-logistic distribution. In this chapter, we will study recently introduced shape parameter estimators. Different properties of the proposed estimators were obtained and their efficiency was evaluated for finite sample sizes using simulation results.

5.1. Introduction

The log-logistic distribution is obtained through the logarithmic transformation of the logistic distribution. This model is widely used in different fields of study. It has been used for modeling failure data, lifetimes and insurance claims, among others. A random variable X has the classic or two-parameter log-logistic distribution if its distribution function (d.f.) is given by

[5.1]

with α the shape parameter and σ the scale parameter. In the economic field, this probabilistic model is also known as the Fisk (1961) distribution. This is a Pareto-type distribution with the tail (or Pareto) index equal to α (Ahsanullah and Alzaatreh 2018). More precisely, we can write

[5.2]

with l(x) a slowly varying function at infinity that measures the departure of F, in [5.1], to ...